Notation for Maps between Topological Spaces

[Rasalhague]

I'm used to the notation f : X --> Y for a map, where X and Y are sets. I recently came across this notation for a map between topological spaces, where the second item of each pair is a topology on the first:

f : (X,{t}_a) --> (Y,{t_b})

Is the notation to be read "f maps each element of X to an element of Y, and f also maps each element, of {t}_a to an element of {t_b}? (Presumably the domain and codomain aren't to be understood as in the nested sets definition of a tuple.)

Source: Fecko: Differential Geometry and Lie Groups for Physicists.

 

[Hurkyl]

The notation is to be read "f is a map from the topological space (X,{t}_a) to the topological space (Y,{t_b})".

What it means to be such a map depends on context. The standard meaning is that it is a map of the underlying sets that has the property of being continuous.

 

[Rasalhague]

Thanks Hurkyl. The example was indeed about continuous maps.